LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOV 2006

PH 1809 – CLASSICAL MECHANICS

Date & Time : 02-11-2006/1.00-4.00    Dept. No.                                                       Max. : 100 Marks

PART A                      ( 10×2 = 20)

1. What are cyclic coordinates? Show that the momentum conjugate to a cyclic coordinate

is a constant

2. Give an example of a velocity dependent potential
3. State and express Hamilton’s variational principle.
4. What are Euler’s angles?.
5. Show that the kinetic energy T for a torque free motion of a rigid body is

a constant of motion.

6. What is meant by canonical transformation?
7. Show that the generating function F₄ = pP generates a transformation that interchanges

momenta and coordinates.

8. Show that [q,H]_q,p = q dot and [p,H]_q,p = p dot
9. Express the Hamiltonian using Hamilton’s characteristic function W in polar coordinates

for a particle under a central force V(r).

10. Define action variable J and angle variable w.

PART B                     (4×7.5 = 30)

Answer any Four questions only

11a Establish the relation between the Lagrangian and the Hamiltonian   (4 marks).

b.Obtain the equations of motion of a simple pendulum using the Hamiltonian formulation.

(3.5 marks)

12. Obtain Hamilton’s equations of motion from the variational principle.
13. Solve the equation of orbit given : q = l ò   dr/r²   / [2m (E+ V(r) – l²/2mr² ]^½    +   q’

for an attractive central potential and classify the orbits in terms of e and E.

14a Obtain the tranformation equation for the generating function F₂(q,P,t)    (4.5 marks)

b Show that the transformation Q = q +  ip and P = q – iP is not canonical   (3marks)

15. Solve the harmonic oscillator problem by the HJ method.

PART C                     (4×12.5 = 50)

Answer any Four questions only

16a. Derive the general  form of Lagrange’s equation using D’Alembert  principle.  (8 marks)

1. A particle of mass m moves in one dimension such that it has the Lagrangian

L = m²x⁴/12 + mx²V(x) –V²(x) where V is some differentiable function of x. Find the

equation of motion for x.           (4.5 marks)

17a. Obtain Euler’s equations of  motion for rigid body acted upon by a torque N  (6 marks)

1. Solve the Euler’s equation of motion for a symmetric top I₁=I₂ ≠ I₃ with no torque

acting on it                                                                                  (6.5 marks)

18a. Show that the Poisson bracket is invariant under canonical transformation  (8 marks)

1. Prove that an infinitesimal canonical transformation does not change the value of the

Hamiltonian of a system.   (4.5 marks)

19. Solve the Kepler’s problem in action-angle variables.
20. Write notes on any TWO of the following
21. i) Constraints of motion
22. ii) Coriolis Effect

iii)  Hamilton Jacobi method.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2010

    PH 1814 / 1809  – CLASSICAL MECHANICS

Date : 03-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART A

Answer ALL questions                                                                                               (10×2 = 20)

1. What are cyclic coordinates?
2. Prove that d/dt (mT) = where T is the kinetic energy of the particle.
3. State and express Hamilton’s variational principle.
4. Give two differences between the Lagrangian and Hamiltonian methods of obtaining the equations of motion.
5. Using the definition of . Show that .
6. What are Euler’s angles?
7. Show that p = 1/Q and q = PQ² is canonical
8. Show that [p_x,L_z] = -p_y
9. Show that the Hamiltonian is a constant of motion if it is not an explicit function of time.
10. What is Hamilton’s principal function S?

PART B

Answer any FOUR questions                                                                                                (4×7.5 = 30)

11. Show that the charged particle in an electromagnetic field has a potential .
12. Reverse the Legendre’s transformation to derive the properties of L(q,,t) from H(q,p,t) treating the q_i as independent quantities and show that it leads to the Lagrangian equation of motion
13. Set up the Lagrangian for a particle of mass m in a central force using polar coordinates (r, q) and hence obtain the differential equation of orbit in the form: d²u/dq² + u = -m/l² d/du V(1/u)
14. Prove the invariance of Poisson’s brackets under a canonical transformation.
15. Solve by the Hamilton Jacobi method the motion of a particle in a plane under the action of a central force V(r) to obtain the equation of the orbit.

PART C

Answer any FOUR questions                                                                                                (4×12.5 = 50)

16 a. Show that the Lagrange’s equation can be derived from Hamilton’s principle for a conservative holonomic system.                                                                                                                              (6)

6. A particle of mass m moves in one dimension such that it has the Lagrangian
   L=m²/12 + mV(x) – V²(x) where V is some differential function of x. Find the equation of motion for x.                                   (6.5)

17 a. Using the definition of the Hamiltonian H(q,p,t) =  – L(,t) obtain Hamilton’s canonical equations.                                                                                                                            (7.5)

1. The Lagrangian for a system of one degree of freedom is,

L = m/2(sin²wt + qwsin2wt +q²w²). What is the Hamiltonian of the system?                              (5)

18  a.Give an account of the theory of canonical transformations.                                                          (6.5)

b.Obtain the transformation equations for the generating functions F₄(p,P,t).                        (6)

19   a. For the Kepler’s problem in action-angle variables assume the expression for the action integral as J_r = [2mE + 2mk/r – (J_q + J_j )² / 4p²r² ]^1/2.dr. Solve this integral to show that t² µ a³ where t is the time period of any planet with semi-major axis ‘ a ‘ about the Sun.                                                                         (8)

4. A particle moves in periodic motion in one dimension under the influence of a potential V(x) = F|(x)| where F is a constant. Using the action-angle variables find the period of the motion as a function of the particle energy.          (4.5)
5. Write notes on,
6. i) Applications of Coriolis Effect              ii) Infinitésimal canonical transformations

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2012

PH 1817 – CLASSICAL MECHANICS

Date : 01/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions:                                                                                                                     (10×2=20)

1. What is a non- holonomic constraint? Give an example.
2. Show that Newton’s Second law can be obtained from the Lagrange’s equation.
3. What is an inertia tensor?
4. Show that the kinetic energy T for torque free motion of rigid body is a constant of motion.
5. Show that the generating function F₂ = Σq_ip_i generates an identity transformation.
6. Show that [q,H]_q,p = and [p,H] _q,p =
7. Define a canonical transformation.
8. What are the fundamental Poisson’s brackets?
9. Explain the physical significance of Hamilton’s characteristics function S.
10. What is Hamilton’s characteristic function W?

PART – B

Answer any FOUR questions:                                                                                                          (4×7.5=30)

11. Show that the charged particle in an electromagnetic field has a potential U = qf – qA.v
12. Obtain the expression for the Coriolis effect as 2m(w x v_r ) where v_r is the velocity in the rotational frame of reference. State its importance in the Earth related phenomenon.
13. Considering the scattering of charged particles as a central force problem obtain an expression for the scattering cross-section s(Q) ie. Rutherford Scattering formula.
14. Obtain the transformation equations for the generating functions F₁(q,Q,t) and F₂(q,P,t)
15. Using the Hamilton-Jacobi method obtain the equation of orbit for a particle in a plane under the action of a central potential V(r).

PART – C

Answer any FOUR questions:                                                                                                       (4×12.5=50)

16   a) Set up the Lagrangian for a particle of mass m in a central force using polar coordinates
(r, q) and hence obtain the differential equation of orbit of the form:

d²u/dq² + u = -m/l² d/du V(1/u)                                                                                       (7.5)

1. b) A particle of mass m is constrained to move under gravity without friction on the inside of a paraboloid of revolution whose axis is vertical. Write the Lagrangian and find the equation of motion. What is the condition on the particle’s initial velocity to produce circular motion? Find the period of small oscillations about this circular motion.                                       (5)

17   a) Obtain Euler’s equations of motion for the rigid body acted upon by a torque N.    (6)

6. b) Solve the Euler’s equation of motion for a symmetric top I₁ = I₂ ≠ I₃ with no torque acting on it.                                                                                                                                                      (6.5)

18   a) Prove the invariance of Poisson’s brackets under a canonical transformation.        (6.5)

1. b) Prove that the infinitesimal canonical transformation does not change the value of the Hamiltonian. (6)
2. Define action and angle variables. Solve the Kepler’s problem in action-angle variables.
3. Write notes on any TWO of the following
4. i) Euler’s angles ii) Solution of one dimensional oscillator by HJ method

iii) Linear triatomic molecule.

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